54 
SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 
or, to the modulus 
(9) 
8 aK _ 4aK/c' 
r^ + rz \/(^V’3)’ 
= 4KP^sn/K'sn(l-/)K', 
a 
M = -27r(r3-r2)(K-E) 
-47rv/(r2r3)(K-E)-,, 
K 
Q = 27r(l—/)—4Kzn/K'. 
_ _ OE _ OB _ BE 
'^’.3 +^'2 OB OB) BD 
DPB = am/Kh DEP = am (l-/) K', 
dn/K' = 
Bj3 
BD' 
The article in the ‘Trans. American Math. Society,’ October, 1907 (A.M.S.), may 
be consulted for an elaborate and detailed discussion of the elliptic function analysis 
and procedure of former writers, and a numerical calculation is given there for the 
helix employed originally by Viriamu Jones. Measurement of fig. 2 gives k, and 
thence, from Legendre’s tables, K, E, Yxp-, and^ = Fxfy/W. 
Another numerical application of these formulas can be chosen from the dimensions 
of the current weigher at the N.P.L., Teddington, described in ‘ Phil. Trans.,’ 1907. 
14. Integrate Q with respect to b to obtain the magnetic potential of the solenoidal 
current sheet, or of the equivalent close helical winding in the ampere balance. 
In these integrations with respect to h the form Q (MQ) of the III. E.I. comes in 
most appropriate as not involving b in MQ, and then 
( 1 ) 
Q db = 
..,7 T Aa cos 6 + cd b dO db 
MQ^ I^ 
^ 7 f Aa cos d + rP 7 ^ 
2 .?.-J(A«cos 0 + a^)(l+jA.)|| 
M 
27r6-^-Pa-6Q(MQ) 
ZTT 
P(r< + QA + Q 6 . 
This solenoidal magnetic potential is the same as that of the cylinder on which the 
helix is wound, and so is the equivalent of the axial component of the gravitation 
attraction of the solid cylinder, and this is the difference of the potentials of the 
